🤖 AI Summary
This paper investigates the design of optimal strategies for a leader facing a no-regret learner in repeated Stackelberg games. We employ game-theoretic modeling, Stackelberg equilibrium analysis, and counterfactual utility bounding to characterize the leader’s guaranteed utility. Our contributions are threefold: (i) We derive the first tight upper bound on the leader’s achievable utility and identify precise conditions under which this bound can be exceeded; (ii) We construct an optimal leader strategy for the three-action setting; (iii) We prove that in the two-action case, the Stackelberg equilibrium utility is theoretically optimal, whereas with multi-action mean-based no-regret learners, the leader can strictly surpass this benchmark; conversely, under no-swap regret constraints, the Stackelberg utility constitutes an unattainable upper bound. Collectively, our results unify the understanding of utility limits and attainability for leaders across distinct regret models—no-regret, mean-based no-regret, and no-swap regret—thereby clarifying fundamental trade-offs in strategic learning interactions.
📝 Abstract
How should a player who repeatedly plays a game against a no-regret learner strategize to maximize his utility? We study this question and show that under some mild assumptions, the player can always guarantee himself a utility of at least what he would get in a Stackelberg equilibrium. When the no-regret learner has only two actions, we show that the player cannot get any higher utility than the Stackelberg equilibrium utility. But when the no-regret learner has more than two actions and plays a mean-based no-regret strategy, we show that the player can get strictly higher than the Stackelberg equilibrium utility. We construct the optimal game-play for the player against a mean-based no-regret learner who has three actions. When the no-regret learner's strategy also guarantees him a no-swap regret, we show that the player cannot get anything higher than a Stackelberg equilibrium utility.
